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# Mathematics > Combinatorics

# Title: A Tight Approximation Algorithm for the Cluster Vertex Deletion Problem

(Submitted on 16 Jul 2020 (v1), last revised 18 Oct 2021 (this version, v3))

Abstract: We give the first $2$-approximation algorithm for the cluster vertex deletion problem. This is tight, since approximating the problem within any constant factor smaller than $2$ is UGC-hard. Our algorithm combines the previous approaches, based on the local ratio technique and the management of true twins, with a novel construction of a 'good' cost function on the vertices at distance at most $2$ from any vertex of the input graph.

As an additional contribution, we also study cluster vertex deletion from the polyhedral perspective, where we prove almost matching upper and lower bounds on how well linear programming relaxations can approximate the problem.

## Submission history

From: Manuel Aprile [view email]**[v1]**Thu, 16 Jul 2020 01:12:45 GMT (27kb)

**[v2]**Thu, 19 Nov 2020 04:31:47 GMT (31kb)

**[v3]**Mon, 18 Oct 2021 11:41:44 GMT (35kb)

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